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ETH/PSI LECTURES 1 Electromagnetic Fields: Transverse vs. Longitudinal 1 SIMPLE GEOMETRICAL DISTINCTION: Longitudinal Field: Propagation direction parallel to the electric field vector E k Transverse Field: Propagation direction perpendicular to the electric field vector E B k 2 QUESTION TO BE ANSWERED HERE: Why is this important? The transverse/longitudinal distinction rarely seems to arise in practical involvement with electromagnetic devices. 3 THE ANSWER LEADS TO MANY OTHER QUESTIONS: • In application to AMO problems, wavelengths are almost always >> the size of the system: >>R. This suggests the dipole approximation, where transverse/longitudinal seems hardly to matter. Isn’t the dipole approximation always valid in practical environments? • There exists a simple gauge transformation that connects the Ap and rE forms of the interaction. This basically relates transverse and longitudinal fields. How can there be important distinctions? • Gauge transformations are for “ivory tower” theoreticians. Why should they be of any importance in the laboratory? • “Ponderomotive” potential will be shown to be very significant. “Ponderomotive” is a clumsy 19th century word. Why is it so important? • Electromagnetic potentials in general will be mentioned often, but potentials are regarded as only auxiliary quantities secondary to electromagnetic fields. Why are potentials so interesting? • And so on … Answers to these questions will raise other questions. 4 GENERAL STRATEGY Longitudinal and transverse fields will be described in detail to emphasize how different they are. Then it will be shown how it has been possible for two such different entities to become indistinguishable to so many observers. 5 LONGITUDINAL FIELDS Static fields like the Coulomb field or a uniform constant electric field are special cases of longitudinal fields. The simplest case is a static uniform field such as that in a parallel-plate capacitor with a constant potential difference between the plates. 6 LONGITUDINAL FIELDS Static electric fields can be described as the gradient of a scalar potential function . This is measurable in the laboratory as a voltage difference. r E; E The arrows represent either the direction of the electric field or the direction of propagation of the field in the quasistatic-electric (QSE) case. E, k This is a longitudinal field. The field propagates in the direction of E. 7 QUASISTATIC ELECTRIC FIELDS Quasistatic electric (QSE) fields can be completely described by a scalar potential in a simple extension of the static case. (t ) r E t ; E t (t ) QSE fields are longitudinal fields. This is all basically electrostatics; the simplest possible problem in electrodynamics. 8 MAXWELL EQUATIONS IN FREE SPACE Use Gaussian units for clarity. There is no need for ε0 or μ0 E 4 B 0 1 E B 0 c t 1 4 B E J c t c Charge density = ρ ; Current density = J The Maxwell equations apply only to fields; potentials do not enter. Maxwell’s equations must be invariant in a gauge transformation . 9 PROPAGATING FIELDS When there are no sources for a field : = 0 , J = 0 , then the only solution that can exist for the Maxwell equations is a propagating, transverse field. Once formed, a propagating field can continue indefinitely in a vacuum. Propagating fields are characterized by periodic behavior depending on a propagation phase t k r The propagation vector k is in the propagation direction, and has the magnitude k = /c . 10 PROPERTIES OF THE TRANSVERSE FIELD A transverse field must have both electric and magnetic components, and they are of equal magnitude (in Gaussian units). The electric field vector, the magnetic field vector, and the propagation vector form a mutually orthogonal triad. The field propagates with velocity c in vacuum. A transverse field is a fundamentally relativistic phenomenon. For most purposes, laboratory air can be regarded as a vacuum. 11 VECTOR FIELDS Because a transverse field propagates at velocity c, it is relativistic. If it is sufficiently strong, it imposes relativistic behavior on systems with which it interacts. (This remains underappreciated in the AMO community.) In both classical electromagnetic theory and in quantum electrodynamics, the electromagnetic field (or photon field) is a vector field represented by a 4-vector potential: A : ( , A ) consisting of a scalar potential for the time component and the 3-vector potential for the spatial component. Electric and magnetic fields follow from the potentials by the expressions E 1 A, c t B A 12 LORENTZ INVARIANTS (The concepts discussed here will be developed in detail in later lectures.) A Lorentz invariant is a quantity that maintains the same value when any Lorentz transformation is employed. A Lorentz transformation relates quantities in two systems moving at constant velocity with respect to each other. There are two basic Lorentz invariants for electromagnetic fields: E 2 B 2 constant E B constant 13 BASIC FIELD TYPES There are 3 limiting cases of electromagnetic fields: Static electric (longitudinal) Plane wave (transverse) Static magnetic E2 – B2 EB E2 0 0 0 - B2 0 Static electric Quasistatic electric (QSE) Plane wave (PW) short-pulse plane wave; focused plane wave Static magnetic Quasistatic magnetic (QSM) - not of further interest 14 As the fields become stronger, the differences become more important. Electromagnetic fields can hardly be more different than these 3 special cases, and the conclusion is that: A QUASISTATIC ELECTRIC FIELD (A LONGITUDINAL FIELD) IS AS DIFFERENT FROM A PLANE WAVE FIELD (A TRANSVERSE FIELD) AS THE PLANE WAVE FIELD IS FROM A QUASISTATIC MAGNETIC FIELD. How can it be that so little distinction is made in the practical world between longitudinal and transverse fields?? 15 There are two very different reasons for the general lack of awareness of the difference between longitudinal and transverse fields: The existence of the Göppert-Mayer gauge transformation. The special nature of the ponderomotive potential. 16 GAUGE TRANSFORMATIONS This will be a brief review. More details will follow in later lectures. The connection between potentials and fields is 1 E A, B A c t The fields are unchanged by a transformation 1 , A A c t where is a scalar function that must satisfy the homogeneous wave equation 2 1 2 2 2 0. c t 17 THE GÖPPERT-MAYER (GM) GAUGE TRANSFORMATION This widely-used gauge transformation applies to plane-wave fields only in the dipole approximation, where the traveling-wave phase can be replaced by the purely time-dependent approximation: t k r t The initial plane-wave potentials are 0, A A (t ) The GM transformation function is GM r A (t ) The transformed potentials are GM r E (t ), A GM 0 The GM gauge transformation replaces a pure vector potential with a pure scalar potential. 18 • • • • PROPERTIES OF THE GM GAUGE GM gauge often called “length gauge” because = - r·E(t) . GM gauge is much favored because a scalar potential is much simpler for analysis than a vector potential. A substantial part of the AMO community regard the GM gauge as “the fundamental gauge”. This view of the GM gauge exists also in the strong-field community, where it can be catastrophic. Two (at least) major problems exist with the GM gauge: 1. It is not possible to represent fully a transverse field, fundamentally a vector field, with a scalar potential. 2. The generating function GM does NOT satisfy the homogeneous wave equation. This has consequences. 19 MAXWELL EQUATIONS Free space, Gaussian units E 4 B 0 1 E B 0 c t 1 4 B E J c t c PLANE WAVE; ρ=0, J=0 LENGTH GAUGE; ρ=0, J0 E 0 B 0 1 E B 0 c t 1 B E 0 c t E E (t ) Ε 0, Ε 0 B 0 E 4J t 20 A BASIC FLAW Maxwell’s equations depend only on fields, not potentials. Maxwell’s equations should be invariant in a gauge transformation. The GM gauge (length gauge) violates this basic fact. (This has apparently escaped notice in the user community.) 21 GM GAUGE IMPLIES AN EXTERNAL SOURCE The fact that the gauge transformation function does not satisfy the homogeneous wave equation is coupled to the fact that Maxwell’s equations require an external source for the length gauge. This means that external energy can be pumped into a system, unlike the plane wave where no sources exist. This can cause problems for the “Simpleman Method”, which is based on the length gauge. When tracked for more than a wave period, the “hidden source” of the length gauge can make enough of a spurious input that serious mistakes can happen. A major example is the path of a photoelectron detached by a laser field. “Simpleman” predicts that there is no HHG with circular polarization because the electron “walks away” from the ion. 22 SIMPLICITY OF “SIMPLEMAN” The Simpleman method assumes that atomic ionization occurs by tunneling through a potential barrier. This view is possible only with the scalar r·E potential. Tunneling involves the assumption that the photoelectron starts at the outer edge of the tunnel with zero velocity, and is then accelerated by the external field, treated as a QSE field. This requires the assumption of an external source of energy to “drive” the photoelectron. This can lead to completely unphysical predictions. An example, shown next, is the prediction for the motion of a photoelectron produced by a circularly polarized field. 23 UNPHYSICAL LENGTH-GAUGE BEHAVIOR WITH CIRCULAR POLARIZATION Displacement in initial E direction (a) atom Displacement in initial B direction Angular momentum (a.u.) (b) 5000 3000 1000 -1000 -3000 -5000 0 20 t 40 60 THE PROBLEM Each photon absorbed from a circularly polarized field adds one quantum unit of angular momentum to the photoelectron, as measured from the center of the atom. The Simpleman “walkaway” starts at zero angular momentum, and after ten cycles, has acquired about 5000 quantum units of angular momentum. This is supplied by the non-existent external current, not by laser photons. A circularly polarized laser can impart only one sign of angular momentum to a system; it cannot provide an oscillating sense of angular momentum as predicted by Simpleman. 25 FURTHER REMARKS ABOUT GAUGES Description of a plane-wave field by a vector potential alone, with = 0 is called the Coulomb gauge, or radiation gauge. When the dipole approximation is applied, the Coulomb gauge is often called the velocity gauge. The GM gauge (length gauge) is gauge-equivalent only to the velocity gauge, it is NOT gauge-equivalent to a full representation of a plane wave. The dipole approximation has validity within a domain limited at both high frequencies and low frequencies. The GM gauge is also limited at both high frequencies and low frequencies. 26 27 To the present day, very few AMO physicists are aware that there is a lower frequency limit to the applicability of the dipole approximation. The conviction remains pervasive that the dipole approximation requires only that the wavelength be much larger than the size of the system with which it interacts. This gives an upper frequency limit to the validity of the dipole approximation. The dipole approximation is NOT VALID for the RFBeta problem. 28 PONDEROMOTIVE POTENTIAL The ponderomotive energy or, more specifically, ponderomotive potential, of a particle of charge q and mass m in a plane-wave field is: q2 Up A2 2m Where the angle brackets indicate the cycle average of the squared vector potential. A relativistic treatment of a charged particle in a plane-wave field gives Up as determining the zero-point energy. It is the potential energy of a charged particle in the field. For example, even nonrelativistically, if an atom is ionized by a strong laser field, the threshold energy for ionization is not just the binding energy EB , but it is the sum of EB and Up . 29 TRADITIONAL VIEW OF UP It was recognized in the 19th century (1885) that, in a nonuniform electromagnetic field, a charged particle is subjected to a ponderomotive force that acts to move the particle from regions of high field intensity to regions of lower intensity. In a focused laser beam, ponderomotive forces can be very large. They act to expel a charged particle from the beam. This led to a basic experiment that showed an external electron beam being scattered from a focused laser beam. [Bucksbaum et al., PRL 58, 349 (1987)] 30 31 MODERN VIEW OF PONDEROMOTIVE POTENTIAL Relativistic calculations for the properties of an electron in a non-perturbatively strong field shows that the mass shell for the electron is altered by Up : p p m ( p nk )( p nk ) m m 2 2 2 m 2mU p 2 in units with ħ=1, c=1; k = propagation 4-vector; n = integer. HRR, J Math Phys 3, 57 (1962); 3, 387 (1962). Nikishov and Ritus, Sov Phys – JETP 19, 529 (1964). Brown and Kibble, Phys Rev 133, A705 (1964). HRR and Eberly, Phys Rev 151, 1058 (1966). 32 Up IS A TRUE POTENTIAL ENERGY If an electron is ionized at threshold: • Input of energy 𝐸𝐵 + 𝑈𝑝 is required • The photoelectron has zero kinetic energy at threshold • If the pulse lasts long enough for the electron to reach the edge of the laser beam, it will have been accelerated to kinetic energy 𝑈𝑝 . • This is directly a conversion of potential to kinetic energy. • Confusion comes from the common description of 𝑈𝑝 as a “quiver energy”, implying classical motion with energy 𝑈𝑝 at threshold. 33 Up DOES NOT EXIST IN THE LENGTH GAUGE In the transformation from the velocity gauge to the length gauge, A2 disappears altogether. Since 𝑈𝑝 ~ 𝐴2 , this means that there is no ponderomotive potential term that can be identified in the length gauge. It is the ponderomotive potential that makes possible judgments about magnetic field effects and relativistic effects. Those judgments cannot be made in the length gauge. Because the potential in the length gauge is of the form -r·E, everything appears to depend solely on the electric field. 34 Up AS A MEASURE OF PLANE-WAVE PHENOMENA Onset of relativistic effects: When Up = (mc2), then relativistic behavior must occur. Failure of the dipole approximation: • Upper limit on frequency (lower limit on wavelength): Dipole approximation requires >> size of bound system. • Lower limit on frequency (upper limit on wavelength): Dipole approximation limited by magnetically-caused displacement of the electron path from a simple oscillation of electric-field effects, or 0 <1, I < 8c3 a.u. This comes from the figure-8 nature of electron motion in a plane-wave field. 35 FIGURE-8 MOTION OF A FREE ELECTRON IN A PLANE-WAVE FIELD k E 0 c 2z f 1 z f c 2z f , zf 2U p mc 2 c zf I 0 4(1 z f ) 4 8c 3 c 0 1 zf I 8c 3 Strong fields and/or low frequencies invalidate the dipole approximation. 36 QUALITATIVE BEHAVIOR IN LENGTH GAUGE Wavelength (nm) 103 102 10 104 103 10 m CO2 101 100 eV 800 nm Ti:sapph 1020 STRONG FIELD 102 1019 Intensity (a.u.) 1018 101 1017 Electric field = 1 a.u. 100 1016 10-1 1015 10-2 Path to = 0 WEAK FIELD -3 10 1014 1013 10-4 10-3 10-2 10-1 Field frequency (a.u.) 100 101 Intensity (W/cm2) 4 QUALITATIVE BEHAVIOR IN COULOMB GAUGE Wavelength (nm) 103 102 10 10 m CO2 cts e ff RELATIVISTIC iel d ag ne (2 1016 M = 1 1 1017 NONDIPOLE 10-2 DIPOLE 1015 0 10-1 zf = m 1018 tic f Up 100 = ) la e R ti s i tiv Intensity (a.u.) 2 c 1019 ce 102 101 1020 cts 103 100 eV 800 nm Ti:sapph ef fe 104 101 Path to =0 10-3 1014 1013 10-4 10-3 10-2 10-1 Field frequency (a.u.) 100 101 Intensity (W/cm2) 4 It is conventional to regard the electric field as the dominant influence on a charged particle even when (as in a plane-wave field) the magnetic field is of comparable magnitude. The reason is that only electric fields can transfer energy to a charged particle. v Lorentz F q E B Work F C c Lorentz ds qE ds Only the electric field can do work on a charged particle (i.e. transfer energy) because the force exerted by the magnetic field is always perpendicular to the direction of motion: (𝒗𝑩) ds . 39 REQUIREMENTS FOR VERY-LOW-FREQUENCY STRONG FIELDS The low frequency limit of the laser domain is at about =10m. The ability of a laser field to transfer energy at this frequency is very limited. For example, it is possible to ionize xenon, but helium has never been ionized at this wavelength. Energy transfer requires a significant electric field, but the electric field from available sources is very small at this low frequency. If energy transfer is not required, but only a large ponderomotive energy, large ponderomotive energies are possible down to extremely low frequencies. 40 Figure 2 101 100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 106 1021 Ti105 20 4 100 eV 10 CO Sapph 2 10 3 1019 10 1018 Relativistic 102 1017 Usual high101 Domain 1016 100 intensity domain 1015 10-1 2) c 1014 10-2 m = 1013 p 10-3 U 12 (2 10 10-4 1 = 10-5 1011 f z 10-6 1010 10-7 109 8 10-8 Nonrelativistic 10 10-9 107 -10 6 10 10 SOI New high10-11 105 -12 UA SOI: Standard Oil of Indiana experiment, 1981 10 intensity domain 104 -13 10 UA: University of Arizona experiment, 1984 103 -14 10 102 -15 10 100 101 102 103 104 105 106 107 108 109 1010 1011 field frequency (MHz) Intensity (W/cm2) Intensity (a.u.) 102 Wavelength (m) 41 There is a large gap between the laser domain and the AM radio domain (microwaves, high frequency rf) where existing technology cannot produce the necessary Up = O(mc2). Below the laser domain, no thought is ever given to singleelectron processes. Everything seems to be completely classical. Only currents of large numbers of electrons need to be considered. Our PSI experiment is entirely novel: It is designed to act on the quantum properties of a single electron in an otherwise classical domain. 42 MIXED TRANSVERSE AND LONGITUDINAL FIELDS For very low frequencies, pair production is so improbable that the electromagnetic field is linear: the sum of any two solutions of Maxwell’s equations is also a solution of Maxwell’s equations. Example: a transverse field impinging on an atom will induce an opposing field created by a change in the nucleus-(atomic electrons) distance. The opposing field is purely QSE. For a free atom or molecule, the sum of the two fields will result in cancellation of the electric field altogether. The magnetic component of the transverse field remains, but the transverse electromagnetic field is essentially removed. Magnetic fields are unaffected by atomic electron influences. Practical evidence: NMR (nuclear magnetic resonance; also known as MRI for “magnetic resonance imaging”) examines nuclear properties by application of an external magnetic field. 43 In a positive ion, the atomic electrons cannot completely cancel an applied transverse electromagnetic field. The remaining electric field will still maintain the appropriate phasing with the original (unaffected) magnetic field. A remnant of the transverse field thus will exist, with its amplitude determined by the size of the remaining electric field. There will be a surplus magnetic field that is incapable of producing transverse-field effects. Consequence: Induced longitudinal fields can reduce the amplitude of a transverse field, but cannot add to it because of the lack of a suitable magnetic component. 44 SUMMARY (page 1) Strong longitudinal fields have the Lorentz invariant E2 – B2 >> 0; strong magnetic fields have E2 – B2 << 0. Plane-wave fields always have E2 – B2 = 0. Transverse and longitudinal electric fields are as different as possible. When the dipole approximation is valid, there exists a gauge transformation (the GM transformation) that connects a dipoleapproximation transverse field (velocity gauge) to a longitudinal field (length gauge). The GM gauge transformation function does not satisfy the required homogeneous wave equation. It is anomalous. The gauge-transformed problem does not satisfy the same Maxwell equations as the original problem. This is a significant pathology, since the Maxwell equations involve only the fields, not the potentials. The domain of validity of the dipole approximation is limited at both low and high frequencies. The AMO community knows only the high 45 frequency limit. SUMMARY (page 2) The wrong Maxwell equations in the length gauge are associated with a spurious external source that does not exist in the velocity gauge. This spurious source can introduce massive errors into the problem if it is tracked for a long enough time. The ponderomotive potential is a major qualitative factor in strongfield problems, but it is hidden in the length gauge. That is, longitudinal fields do not exhibit a ponderomotive potential. Description of the ponderomotive potential as a “quiver energy” is misleading because the classical “quiver” does not arise until after a detached electron leaves the beam. Ponderomotive potentials can be relativistic even at extremely low field frequencies. This is not usually apparent because extremely low frequencies do not transfer energy effectively. Classical circuits involve large numbers of electrons, usually confined to conductors, where their behavior is analogous to the behavior in 46 longitudinal fields. SUMMARY (page 3) The PSI experiments are specially designed to reveal relativistic behavior of a single electron at super-low frequencies. This is completely novel. Note: The problem with the GM gauge is not an isolated problem. Future lectures will return to the subject of gauges, since neither textbooks nor current literature exhibit the difficulties that can arise. 47